Microsoft PowerPoint - DFT.ppt [Compatibility Mode]Nuno Vasconcelos UCSD
The Discrete-Space Fourier Transform • as in 1D, an important concept in linear system analysis
is that of the Fourier transform • the Discrete-Space Fourier Transform is the 2D
extension of the Discrete-Time Fourier Transform 2211][)( ωω ωω eennxX njnj∑∑ −−
2121
2211
2211
2121221 ),( )2(
],[ ωωωω π
ddeeXnnx ∫∫=
q y – inconvenient to evaluate numerically in DSP hardware – we need a discrete version
this is the 2D Discrete Fourier Transform (2D DFT)
2
• before that we consider the sampling problem
Sampling in 2D
• consider an analog signal xc(t1,t2) and let its analog Fourier transform be X (1 2)Fourier transform be Xc(1,2) – we use capital to emphasize that this is analog frequency
• sample with period (T1,T2) to obtain a discrete-space signal
222111 ;2121 ),(],[ TntTntc ttxnnx
Sampling in 2D
∑ ∑ ∞ ∞


−∞= −∞= 1 2 2121 r r TTTT
“discrete spectrum” “analog spectrum”
• Discrete Space spectrum is sum of replicas of analog• Discrete Space spectrum is sum of replicas of analog spectrum – in the “base replica” the analog frequency 1 (2) is mapped
i t th di it l f T ( T ) 4
into the digital frequency 1T1 (2T2) – discrete spectrum has periodicity (2π,2π)
For example 2
ω1
2π⇔
is the critical sampling frequency
• below it we have aliasing
ω2
aliasing
ω1
• this is just like the 1D case, but now there are morenow there are more possibilities for overlap
6
Reconstruction • if there is no aliasing we can recover the signal in a way
similar to the 1D case
( ) ( ) ∑ ∑ ∞ ∞
−− =
( ) ( ) ∑ ∑ −∞= −∞= −−1 2
222 2
111 1
ππ
• note: in 2D there are many more possibilities than in 1D – e.g. the sampling grid does not have to be rectangular, e.g.
hexagonal sampling when T2 = T1/sqrt(3) and

= ==
21 ,),(
7– in practice, however, one usually adopts the rectangular grid
• a sequence of images obtained by down- sampling without any filteringg
• aliasing: the low- frequency parts are
li t d th h t th 8
replicated throughout the low-res image
The role of smoothing
l ta lot
9 • too little leads to aliasing • too much leads to loss of information
Aliasing in video • video frames are the result of temporal sampling
– fast moving objects are above the critical frequency – above a certain speed they are aliased and appear to move
backwards – this was common in old western movies and become known as
the “wagon wheel” effect – here is an example: super-resolution increases the frame rate
and eliminates aliasing
from “Space-Time Resolution in Video” by y E. Shechtman, Y. Caspi and M. Irani
(PAMI 2005).
10
2D-DFT • the 2D-DFT is obtained by sampling the DSFT at regular
frequency intervals
2 2
21 1
1 2,22121 ),(],[
k N
k N
== =
• this turns out to make the 2D-DFT somewhat harder to work with than the DSFT
it i th i 1D– it is the same as in 1D – you might remember that the inverse transform of the product of
two DFTs is not the convolution of the associated signals – but, instead, the “circular convolution” – where does this come from?
• it is better understood by first considering the 2D
11
it is better understood by first considering the 2D Discrete Fourier Series (2D-DFS)
[ ] [ ] [ ]
−−= 1 2
21 212121 ],[),(
n n
jnjn rrnnxrrX
• makes no sense for a periodic signal – the sum will be infinite for any pair r1,r2
ith th 2D DSFT th Z t f ill k h
12
– neither the 2D DSFT or the Z-transform will work here
2D-DFS • the 2D-DFS solves this problem • it is based on the observation that
– any periodic sequence can be represented as a weighted sum of complex exponentials of the form
22 2211 nkjnkj ππ
10 22 ≤≤ Nk
22 kk ππ
is an orthonormal basis of the space of periodic sequences
10 ,10 , 2211
– is an orthonormal basis of the space of periodic sequences
2D-DFS • the 2D-DFS relates x[n1,n2] and X[k1,k2]
( ) [ ] 22111 2 221 1 nkjnkjN N ππ
−−− −
1010 ≤≤≤≤ NkNk • like the DSFT,
properties of the 2D DFS are identical to those of the 1D DFS
10 ,10 2211 −≤≤−≤≤ NkNk
14
– properties of the 2D-DFS are identical to those of the 1D-DFS – with the straightforward extension of separability
Periodic convolution • like the Fourier transform,
– the inverse transform of multiplication is convolution
[ ] [ ] ( ) ( )2121
×↔∗
– however, we have to be careful about how we define convolution – since the sequences have no end, the standard definition
],[],[],[ 22112121 1 2
knknhkkxnny k k
−−= ∑ ∑ ∞
−∞=
∞
−∞=
makes no sense – e.g. if x and h are both positive sequences, this will allways be
infinite
15
infinite
Periodic convolution • to deal with this, we introduce the idea of periodic
convolution • instead of the regular definition
],[],[* 221121 knknhkkxyx −−= ∑ ∑ ∞ ∞
• which, from now on, we refer to as linear convolution
],[],[ 221121 1 2
y k k ∑ ∑
1 11 2N N− −
],[ ],[ 2211
1
0
1
• the only difference is in the summation limits
Periodic convolution • this is simple, but produces a convolution which is
substantially different • let’s go back to our example, now assuming that the
sequences have period (N1=3,N2=2) n2 n2
(1) (2)* (3) (4)
• as before we need four steps ],[ 21 nnx ],[ 21 nnh
17
• as before, we need four steps
Periodic convolution • step 1): express sequences in terms of (k1, k2), and
consider one period only
],[ 21 kkx ],[ 21 kkh
Periodic convolution • step 2): invert h(k1, k2)
1 11 2N N− −
19
Periodic convolution • step 3): shift g(k1, k2) by (n1, n2)
1 11 2N N− −
k1
k2k2
k
(3)(4)
n1
k1
th t t ld b t i t i lti l th t
],[ 21 kkx ],[
=−−
the next step would be to point-wise multiply the two signals and sum
][][ 1 11 2
kkhkkh N N
o
Periodic convolution • this is where we depart from linear convolution • remember that the sequences are periodic, and we really
only care about what happens in the fundamental periodonly care about what happens in the fundamental period • we use the periodicity to fill the values missing in the
flipped sequence k2
22
Periodic convolution • step 4): we can finally point-wise multiply the two signals
and sum
3x1+1x1=4
],[ 21 kkx ],[ 21 nny],[ 2211
Periodic convolution • note: the sequence that results from the convolution is
also periodic • it is important to keep in mind what we have done• it is important to keep in mind what we have done
– we work with a single period (the fundamental period) to make things manageable b t b th t h i di– but remember that we have periodic sequences
– it is like if we were peeking through a window – if we shift, or flip the sequence we need to remember that – the sequence does not simply move out of the window, but the
next period walks in!!! – note, that this
k2 k2
24
Discrete Fourier Transform • all of this is interesting,
– but why do I care about periodic sequences? – all images are finite, I could never have such a sequence
• while this is true – the DFS is the easiest route to learn about the discrete Fourierthe DFS is the easiest route to learn about the discrete Fourier
transform (DFT)
• recall that the DFT is obtained by sampling the DSFT
2 2
21 1
1 2,22121 ),(],[
k N
k N
== =
– we know that when we sample in space we have aliasing in frequency
– well, the same happens when we sample in frequency: we get
25
Discrete Fourier Transform • this means that
– even if we have a finite sequence – when we compute the DFT we are effectively working with a
periodic sequence
• you may recall from 1D signal processing thaty y g p g – when you multiply two DFTs, you do not get convolution in time – but instead something called circular convolution
this is strange: when I flip a signal (e g convolution) it wraps– this is strange: when I flip a signal (e.g. convolution) it wraps around the sequence borders
– well, in 2D it gets much stranger
th l I k h t d t d thi i t• the only way I know how to understand this is to – think about the underlying periodic sequence – establish a connection between the DFT and the DFS of that
26
sequence – use what we have seen for the DFS to help me out with the DFT
Discrete Fourier Transform • let’s start by the relation between DFT and DFS • the DFT is defined as
2 2
21 1
1 2,22121 ),(],[
k N
k N
[ ] <≤ ∑∑
22111 2 ππ
[ ] <≤ ∑∑
22111 2 ππ
jnk N
jN N
kkX ππ
110 Nk <≤ 110 Nn <≤
11
<≤ Nn0 1 11
29
[ ] [ ] [ ]212121 ,,, 21
kkRkkXkkX NN ×=
that working in the DFT domain is equivalent to – working in the DFS domain
[ ] [ ] [ ]212121 21 NN [ ] [ ] [ ]212121 21 NN
– extracting the fundamental period at the end
• we can summarize this as
[ ] [ ] [ ] [ ]2121
2121 ,
↔← →
• in this way, I can work with the DFT without having to worry about aliasing
eperiodiciztruncate
30
↔ →
• this trick can be used to derive all the DFT properties
[ ] [ ] [ ] [ ]21212121 ,,,, kkXkkXnnxnnx eperiodiciztruncate
←↔←
• e.g. what is the inverse transform of a phase shift? – let’s follow the steps
22 kk ππ
– 1) periodicize: this causes the same phase shift in the DFS
[ ] [ ] 22 2
11 1
← →
↔← →
– 2) compute the inverse DFS: it follows from the properties of the DFS (page 142 on Lim) that we get a shift in space
eperiodiciztruncate ←←
DFS (page 142 on Lim) that we get a shift in space
[ ] [ ]221121 ,, mnmnxnny −−=
[ ] [ ] [ ]
– in summary, the new sequence is obtained by making the
[ ] [ ] [ ]21221121 ,,, 21
nnRmnmnxnny NN ×−−=
32
y q y g original periodic, shifting, and taking the fundamental period
Example k2 k2
33
• note that what leaves on one end, enters on the other
Example • for this reason it is called a circular shift
k2
k1
• note that this is way more complicated than in 1D • to get it right we really have to think in terms of the
i di t i f thperiodic extension of the sequence • we will see that this shows up in the properties of the
DFT,
34
35